A Course in Homological Algebra

This book arose out of a course of lectures given at the Swiss Federal Institute of Technology (ETH), Zürich, in 1966–67. The course was first set down as a set of lecture notes, and, in 1968, Professor Eckmann persuaded the authors to build a graduate text out of the notes, taking account, where appropriate, of recent developments in the subject.

The algebraic categories with which we shall be principally concerned in this book are categories of modules over a fixed (unitary) ring Λ and module-homomorphisms. Thus we devote this chapter to a preliminary discussion of Λ-modules.

In Chapter I we discussed various algebraic structures (rings, abelian groups, modules) and their appropriate transformations (homomorphisms). We also saw how certain constructions (for example, the formation of Hom_Λ(A, B) for given Λ-modules A, B) produced new structures out of given structures. Over and above this we introduced certain “universal” constructions (direct sum, direct product) and suggested that they constituted special cases of a general, and important, procedure. Our objective in this chapter is to establish the appropriate mathematical language for the general description of mathematical systems and of mappings of systems, insofar as that language is applicable to homological algebra.

In studying modules, as in studying any algebraic structures, the standard procedure is to look at submodules and associated quotient modules. The extension problem then appears quite naturally: given modules A, B (over a fixed ring Λ) what modules E may be constructed with submodule B and associated quotient module A? The set of equivalence classes of such modules E, written E(A, B) may then be given an abelian group structure in a way first described by Baer [3]. It turns out that this group E(A, B) is naturally isomorphic to a group Ext_Λ(A, B) obtained from A and B by the characteristic, indeed prototypical, methods of homological algebra. To be precise, Ext_Λ(A, B) is the value of the first right derived functor of Hom_Λ(−, B) on the module A, in the sense of Chapter IV.

In this chapter we go to the heart of homological algebra. Everything up to this point can be regarded as providing essential background for the theory of derived functors, and introducing the special cases of Ext_Λ(A, B), Tor^Λ(A, B). Subsequent chapters take up more sophisticated properties of derived functors and special features of the theory in various contexts* (cohomology of groups, cohomology of Lie algebras).

The Künneth formula has its historic origin in algebraic topology. Given two topological spaces X and Y, we may ask how the (singular) homology groups of their topological product X × Y is related to the homology groups of X and Y. This question may be answered by separating the problem into two parts. If C(X), C(Y) C(X × Y) stand for the singular chain complexes of X, Y, X × Y respectively, then a theorem due to Eilenberg-Zilber establishes that the chain complex C(X × Y) is canonically homotopy-equivalent to the tensor product of the chain complexes C(X) and C(Y), ; (for the precise definition of the tensor product of two chain complexes, see Section 1, Example (a)). Thus the problem is reduced to the purely algebraic problem of relating the homology groups of the tensor product of C(X) and C(Y) to the homology groups of C(X) and C(Y). This relation is furnished by the Künneth formula, whose validity we establish under much more general circumstances than would be required by the topological situation. For, in that case, we are concerned with free chain complexes of ℤ-modules; the argument we give permits arbitrary chain complexes C, D of Λ-modules, where Λ is any p.i.d., provided only that one of C, D is flat. This generality allows us then to subsume under the same theory not only the Künneth formula in its original context but also another important result drawn from algebraic topology, the universal coefficient theorem in homology.

In this chapter we shall apply the theory of derived functors to the important special case where the ground ring Λ is the group ring ℤG of an abstract group G over the integers. This will lead us to a definition of cohomology groups Hⁿ(G, A) and homology groups H_n(G, B) n ≧ 0, where A is a left and B a right G-module (we speak of “G-modules” instead of “ℤG-modules”). In developing the theory we shall attempt to deduce as much as possible from general properties of derived functors. Thus, for example, we shall give a proof of the fact that H²(G, A) classifies extensions which is not based on a particular (i.e. standard) resolution.

In this Chapter we shall give a further application of the theory of derived functors. Starting with a Lie algebra g over the field K, we pass to the universal enveloping algebra U_g and define cohomology groups Hⁿ(g, A) for every (left) g-module A, by regarding A as a U_g-module. In Sections 1 through 4 we will proceed in a way parallel to that adopted in Chapter VI in presenting the cohomology theory of groups. We therefore allow ourselves in those sections to leave most of the proofs to the reader. Since our primary concern is with the homological aspects of Lie algebra theory, we will not give proofs of two deep results of Lie algebra theory although they are fundamental for the development of the cohomology theory of Lie algebras; namely, we shall not give a proof for the Birkhoff-Witt Theorem (Theorem 1.2) nor of Theorem 5.2 which says that the bilinear form of certain representations of semi-simple Lie algebras is non-degenerate. Proofs of both results are easily accessible in the literature.

In this chapter we develop the theory of spectral sequences: applications will be found in Section 9 and in Chapter IX. Our procedure will be to base the theory on the study of exact couples, but we do not claim, of course, that this is the unique way to present the theory; indeed, an alternative approach is to be found e.g. in [7]. Spectral sequences themselves frequently arise from filtered differential objects in an abelian category — for example, filtered chain complexes. In such cases it is naturally quite possible to pass directly from the filtered differential object to the spectral sequence without the intervention of the exact couple. However, we believe that the explicit study of the exact couple illuminates the nature of the spectral sequence and of its limit.

In Chapters VI and VII we gave “concrete” applications of the theory of derived functors established in Chapter IV, namely to the category of groups and the category of Lie algebras over a field K. In this chapter our first purpose is to broaden the setting in which a theory of derived functors may be developed. This more general theory is called relative homological algebra, the relativization consisting of replacing the class of all epimorphisms (monomorphisms) by a suitable subclass in defining the notion of projective (injective) object. An important example of such a relativization, which we discuss explicitly, consists in taking, as our projective class of epimorphisms in the category ###m_Λ of Λ-modules, those epimorphisms which split as abelian group homomorphisms.

The first section of this chapter describes how homological algebra arose by abstraction from algebraic topology and how it has contributed to the knowledge of topology. The other four sections describe applications of the methods and results of homological algebra to other parts of algebra. Most of the material presented in these four sections was not available when this text was first published. Since then homological algebra has indeed found a large number of applications in many different fields, ranging from finite and infinite group theory to representation theory, number theory, algebraic topology, and sheaf theory. Today it is a truly indispensable tool in all these fields. For the purpose of illustrating to the reader the range and depth of these developments, we have selected a number of different topics and describe some of the main applications and results. Naturally, the treatments are somewhat cursory, the intention being to give the flavor of the homological methods rather than the details of the arguments and results.